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On the identification of the nonlinearity parameter in the Westervelt equation from boundary measurements
1. | Department of Mathematics, Alpen-Adria-Universität Klagenfurt, 9020 Klagenfurt, Austria |
2. | Department of Mathematics, Texas A&M University, Texas 77843, USA |
We consider an undetermined coefficient inverse problem for a nonlinear partial differential equation occurring in high intensity ultrasound propagation as used in acoustic tomography. In particular, we investigate the recovery of the nonlinearity coefficient commonly labeled as $ B/A $ in the literature which is part of a space dependent coefficient $ \kappa $ in the Westervelt equation governing nonlinear acoustics. Corresponding to the typical measurement setup, the overposed data consists of time trace measurements on some zero or one dimensional set $ \Sigma $ representing the receiving transducer array. After an analysis of the map from $ \kappa $ to the overposed data, we show injectivity of its linearisation and use this as motivation for several iterative schemes to recover $ \kappa $. Numerical simulations will also be shown to illustrate the efficiency of the methods.
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A. B. Bakushinskiĭ,
On a convergence problem of the iterative-regularised Gauss-Newton method, Comput. Math. Math. Phys., 32 (1992), 1353-1359.
|
[2] |
A. B. Bakushinskii,
Remarks on choosing a regularisation parameter using the quasi-optimality and ratio criterion, USSR Comput. Math. Math. Phys., 24 (1984), 181-182.
doi: 10.1016/0041-5553(84)90253-2. |
[3] |
L. Bjørnø,
Characterization of biological media by means of their non-linearity, Ultrasonics, 24 (1986), 254-259.
doi: 10.1016/0041-624x(86)90102-2. |
[4] |
D. T. Blackstock, Approximate equations governing finite-amplitude sound in thermoviscous fluids, Tech Report, GD/E Report, GD-1463-52, General Dynamics Corp., Rochester, NY, 1963. Google Scholar |
[5] |
B. Blaschke, A. Neubauer and O. Scherzer,
On convergence rates for the iteratively regularised Gauss-Newton method, IMA J. Numer. Anal., 17 (1997), 421-436.
doi: 10.1093/imanum/17.3.421. |
[6] |
J. M. Burgers, The Nonlinear Diffusion Equation, Springer, Netherlands, 1974.
doi: 10.1007/978-94-010-1745-9. |
[7] |
V. Burov, I. Gurinovich, O. Rudenko and E. Tagunov, Reconstruction of the spatial distribution of the nonlinearity parameter and sound velocity in acoustic nonlinear tomography, Acoustical Physics, 40 (1994), 816-823. Google Scholar |
[8] |
C. A. Cain, Ultrasonic reflection mode imaging of the nonlinear parameter B/A: A theoretical basis, IEEE 1985 Ultrasonics Symposium, San Francisco, CA, USA, 1985.
doi: 10.1109/ULTSYM.1985.198640. |
[9] |
C. Clason and A. Klassen,
Quasi-solution of linear inverse problems in non-reflexive Banach spaces, J. Inverse Ill-Posed Probl., 26 (2018), 689-702.
doi: 10.1515/jiip-2018-0026. |
[10] |
C. Clason and K. Kunisch,
A duality-based approach to elliptic control problems in non-reflexive Banach spaces, ESAIM Control Optim. Calc. Var., 17 (2011), 243-266.
doi: 10.1051/cocv/2010003. |
[11] |
D. G. Crighton,
Model equations of nonlinear acoustics, Ann. Rev. Fluid Mech., 11 (1979), 11-33.
doi: 10.1146/annurev.fl.11.010179.000303. |
[12] |
H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Mathematics and its Applications, 375, Kluwer Academic Publishers Group, Dordrecht, 1996. |
[13] |
H. W. Engl, K. Kunisch and A. Neubauer,
Convergence rates for Tikhonov regularisation of non-linear ill-posed problems, Inverse Problems, 5 (1989), 523-540.
doi: 10.1088/0266-5611/5/4/007. |
[14] |
L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998. |
[15] |
M. F. Hamilton and D. T. Blackstock, Nonlinear Acoustics, Vol. 1, Academic Press, San Diego, 1998. Google Scholar |
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M. Hanke,
A regularizing Levenberg–Marquardt scheme, with applications to inverse groundwater filtration problems, Inverse Problems, 13 (1997), 79-95.
doi: 10.1088/0266-5611/13/1/007. |
[17] |
M. Hanke, A. Neubauer and O. Scherzer,
A convergence analysis of the Landweber iteration for nonlinear ill-posed problems, Numer. Math., 72 (1995), 21-37.
doi: 10.1007/s002110050158. |
[18] |
F. Hettlich and W. Rundell,
A second degree method for nonlinear inverse problems, SIAM J. Numer. Anal., 37 (2000), 587-620.
doi: 10.1137/S0036142998341246. |
[19] |
B. Hofmann, B. Kaltenbacher, C. Pöschl and O. Scherzer,
A convergence rates result for Tikhonov regularisation in Banach spaces with non-smooth operators, Inverse Problems, 23 (2007), 987-1010.
doi: 10.1088/0266-5611/23/3/009. |
[20] |
T. Hohage,
Logarithmic convergence rates of the iteratively regularised Gauß-Newton method for an inverse potential and an inverse scattering problem, Inverse Problems, 13 (1997), 1279-1299.
doi: 10.1088/0266-5611/13/5/012. |
[21] |
S. Hubmer and R. Ramlau, Nesterov's accelerated gradient method for nonlinear ill-posed problems with a locally convex residual functional, Inverse Problems, 34 (2018), 30pp.
doi: 10.1088/1361-6420/aacebe. |
[22] |
N. Ichida, T. Sato and M. Linzer,
Imaging the nonlinear ultrasonic parameter of a medium, Ultrasonic Imaging, 5 (1983), 295-299.
doi: 10.1177/016173468300500401. |
[23] |
O. Y. Imanuvilov and M. Yamamoto, Carleman estimate and an inverse source problem for the Kelvin-Voigt model for viscoelasticity, Inverse Problems, 35 (2019), 45pp.
doi: 10.1088/1361-6420/ab323e. |
[24] |
V. Isakov, Inverse Problems for Partial Differential Equations, Applied Mathematical Sciences, 127, Springer, New York, 2006.
doi: 10.1007/0-387-32183-7. |
[25] |
V. K. Ivanov,
On linear problems which are not well-posed, Dokl. Akad. Nauk SSSR, 145 (1962), 270-272.
|
[26] |
B. Kaltenbacher,
An iteratively regularized Gauss-Newton-Halley method for solving nonlinear ill-posed problems, Numer. Math., 131 (2015), 33-57.
doi: 10.1007/s00211-014-0682-5. |
[27] |
B. Kaltenbacher,
Mathematics of nonlinear acoustics, Evol. Equ. Control Theory, 4 (2015), 447-491.
doi: 10.3934/eect.2015.4.447. |
[28] |
B. Kaltenbacher, Periodic solutions and multiharmonic expansions for the Westervelt equation, to appear, Evol. Equ. Control Theory.
doi: 10.3934/eect.2020063. |
[29] |
B. Kaltenbacher and I. Lasiecka,
Global existence and exponential decay rates for the Westervelt equation, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 503-523.
doi: 10.3934/dcdss.2009.2.503. |
[30] |
B. Kaltenbacher and A. Klassen, On convergence and convergence rates for Ivanov and Morozov regularisation and application to some parameter identification problems in elliptic PDEs, Inverse Problems, 34 (2018), 24pp.
doi: 10.1088/1361-6420/aab739. |
[31] |
B. Kaltenbacher, A. Neubauer and O.Scherzer, Iterative Regularization Methods for Nonlinear Ill-Posed Problems, Radon Series on Computational and Applied Mathematics, 6, Walter de Gruyter GmbH & Co. KG, Berlin, 2008.
doi: 10.1515/9783110208276. |
[32] |
V. Kuznetsov, Equations of nonlinear acoustics, Soviet Physics - Acoustics, 16 (1971), 467-470. Google Scholar |
[33] |
M. B. Lesser and R. Seebass,
The structure of a weak shock wave undergoing reflexion from a wall, J. Fluid Mech., 31 (1968), 501-528.
doi: 10.1017/S0022112068000303. |
[34] |
M. J. Lighthill, Viscosity effects in sound waves of finite amplitude, in Surveys in Mechanics, Cambridge, at the University Press, 1956, 250–351. |
[35] |
D. Lorenz and N. Worliczek, Necessary conditions for variational regularisation schemes, Inverse Problems, 29 (2013), 19pp.
doi: 10.1088/0266-5611/29/7/075016. |
[36] |
S. Meyer and M. Wilke,
Optimal regularity and long-time behavior of solutions for the Westervelt equation, Appl. Math. Optim., 64 (2011), 257-271.
doi: 10.1007/s00245-011-9138-9. |
[37] |
V. A. Morozov,
On the solution of functional equations by the method of regularisation, Soviet Math. Dokl., 7 (1966), 414-417.
|
[38] |
M. Muhr, V. Nikolić, B. Wohlmuth and L. Wunderlich,
Isogeometric shape optimization for nonlinear ultrasound focusing, Evol. Equ. Control Theory, 8 (2019), 163-202.
doi: 10.3934/eect.2019010. |
[39] |
A. Neubauer,
On Nesterov acceleration for Landweber iteration of linear ill-posed problems, J. Inverse Ill-Posed Probl., 25 (2017), 381-390.
doi: 10.1515/jiip-2016-0060. |
[40] |
A. Neubauer,
Tikhonov-regularisation of ill-posed linear operator equations on closed convex sets, J. Approx. Theory, 53 (1988), 304-320.
doi: 10.1016/0021-9045(88)90025-1. |
[41] |
A. Neubauer and O. Scherzer,
A convergent rate result for a steepest descent method and a minimal error method for the solution of nonlinear ill-posed problems, Z. Anal. Anwendungen, 14 (1995), 369-377.
doi: 10.4171/ZAA/679. |
[42] |
H. Ockendon and J. R. Ockendon, Waves and Compressible Flow, Texts in Applied Mathematics, 47, Springer-Verlag, New York, 2004.
doi: 10.1007/b97537. |
[43] |
A. Pierce,
Unique identification of eigenvalues and coefficients in a parabolic problem, SIAM J. Control Optim., 17 (1979), 494-499.
doi: 10.1137/0317035. |
[44] |
A. Rieder,
On convergence rates of inexact Newton regularizations, Numer. Math., 88 (2001), 347-365.
doi: 10.1007/PL00005448. |
[45] |
W. Rundell and P. E. Sacks,
Reconstruction techniques for classical inverse Sturm-Liouville problems, Math. Comp., 58 (1992), 161-183.
doi: 10.1090/S0025-5718-1992-1106979-0. |
[46] |
O. Scherzer,
A modified Landweber iteration for solving parameter estimation problems, Appl. Math. Optim., 38 (1998), 45-68.
doi: 10.1007/s002459900081. |
[47] |
T. I. Seidman and C. R. Vogel,
Well-posedness and convergence of some regularisation methods for non-linear ill posed problems, Inverse Problems, 5 (1989), 227-238.
doi: 10.1088/0266-5611/5/2/008. |
[48] |
F. Varray, O. Basset, P. Tortoli and C. Cachard,
Extensions of nonlinear B/A parameter imaging methods for echo mode, IEEE Trans. Ultrasonics, Ferroelectrics, and Frequency Control, 58 (2011), 1232-1244.
doi: 10.1109/TUFFC.2011.1933. |
[49] |
P. J. Westervelt,
Parametric acoustic array, J. Acoustical Soc. Amer., 35 (1963), 535-537.
doi: 10.1121/1.1918525. |
[50] |
M. Yamamoto and B. Kaltenbacher, An inverse source problem related to acoustic nonlinearity parameter imaging, to appear, Time-Dependent Problems in Imaging and Parameter Identification, Springer, 2021. Google Scholar |
[51] |
E. A. Zabolotskaya and R. V. Khokhlov, Quasi-plane waves in the non-linear acoustics of confined beams, Soviet Physics - Acoustics, 15 (1969), 35-40. Google Scholar |
[52] |
D. Zhang, X. Chen and X.-F. Gong,
Acoustic nonlinearity parameter tomography for biological tissues via parametric array from a circular piston source - Theoretical analysis and computer simulations, J. Acoustical Soc. Amer., 109 (2001), 1219-1225.
doi: 10.1121/1.1344160. |
[53] |
D. Zhang, X. Gong and S. Ye,
Acoustic nonlinearity parameter tomography for biological specimens via measurements of the second harmonics, J. Acoustical Soc. Amer., 99 (1996), 2397-2402.
doi: 10.1121/1.415427. |
show all references
References:
[1] |
A. B. Bakushinskiĭ,
On a convergence problem of the iterative-regularised Gauss-Newton method, Comput. Math. Math. Phys., 32 (1992), 1353-1359.
|
[2] |
A. B. Bakushinskii,
Remarks on choosing a regularisation parameter using the quasi-optimality and ratio criterion, USSR Comput. Math. Math. Phys., 24 (1984), 181-182.
doi: 10.1016/0041-5553(84)90253-2. |
[3] |
L. Bjørnø,
Characterization of biological media by means of their non-linearity, Ultrasonics, 24 (1986), 254-259.
doi: 10.1016/0041-624x(86)90102-2. |
[4] |
D. T. Blackstock, Approximate equations governing finite-amplitude sound in thermoviscous fluids, Tech Report, GD/E Report, GD-1463-52, General Dynamics Corp., Rochester, NY, 1963. Google Scholar |
[5] |
B. Blaschke, A. Neubauer and O. Scherzer,
On convergence rates for the iteratively regularised Gauss-Newton method, IMA J. Numer. Anal., 17 (1997), 421-436.
doi: 10.1093/imanum/17.3.421. |
[6] |
J. M. Burgers, The Nonlinear Diffusion Equation, Springer, Netherlands, 1974.
doi: 10.1007/978-94-010-1745-9. |
[7] |
V. Burov, I. Gurinovich, O. Rudenko and E. Tagunov, Reconstruction of the spatial distribution of the nonlinearity parameter and sound velocity in acoustic nonlinear tomography, Acoustical Physics, 40 (1994), 816-823. Google Scholar |
[8] |
C. A. Cain, Ultrasonic reflection mode imaging of the nonlinear parameter B/A: A theoretical basis, IEEE 1985 Ultrasonics Symposium, San Francisco, CA, USA, 1985.
doi: 10.1109/ULTSYM.1985.198640. |
[9] |
C. Clason and A. Klassen,
Quasi-solution of linear inverse problems in non-reflexive Banach spaces, J. Inverse Ill-Posed Probl., 26 (2018), 689-702.
doi: 10.1515/jiip-2018-0026. |
[10] |
C. Clason and K. Kunisch,
A duality-based approach to elliptic control problems in non-reflexive Banach spaces, ESAIM Control Optim. Calc. Var., 17 (2011), 243-266.
doi: 10.1051/cocv/2010003. |
[11] |
D. G. Crighton,
Model equations of nonlinear acoustics, Ann. Rev. Fluid Mech., 11 (1979), 11-33.
doi: 10.1146/annurev.fl.11.010179.000303. |
[12] |
H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Mathematics and its Applications, 375, Kluwer Academic Publishers Group, Dordrecht, 1996. |
[13] |
H. W. Engl, K. Kunisch and A. Neubauer,
Convergence rates for Tikhonov regularisation of non-linear ill-posed problems, Inverse Problems, 5 (1989), 523-540.
doi: 10.1088/0266-5611/5/4/007. |
[14] |
L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998. |
[15] |
M. F. Hamilton and D. T. Blackstock, Nonlinear Acoustics, Vol. 1, Academic Press, San Diego, 1998. Google Scholar |
[16] |
M. Hanke,
A regularizing Levenberg–Marquardt scheme, with applications to inverse groundwater filtration problems, Inverse Problems, 13 (1997), 79-95.
doi: 10.1088/0266-5611/13/1/007. |
[17] |
M. Hanke, A. Neubauer and O. Scherzer,
A convergence analysis of the Landweber iteration for nonlinear ill-posed problems, Numer. Math., 72 (1995), 21-37.
doi: 10.1007/s002110050158. |
[18] |
F. Hettlich and W. Rundell,
A second degree method for nonlinear inverse problems, SIAM J. Numer. Anal., 37 (2000), 587-620.
doi: 10.1137/S0036142998341246. |
[19] |
B. Hofmann, B. Kaltenbacher, C. Pöschl and O. Scherzer,
A convergence rates result for Tikhonov regularisation in Banach spaces with non-smooth operators, Inverse Problems, 23 (2007), 987-1010.
doi: 10.1088/0266-5611/23/3/009. |
[20] |
T. Hohage,
Logarithmic convergence rates of the iteratively regularised Gauß-Newton method for an inverse potential and an inverse scattering problem, Inverse Problems, 13 (1997), 1279-1299.
doi: 10.1088/0266-5611/13/5/012. |
[21] |
S. Hubmer and R. Ramlau, Nesterov's accelerated gradient method for nonlinear ill-posed problems with a locally convex residual functional, Inverse Problems, 34 (2018), 30pp.
doi: 10.1088/1361-6420/aacebe. |
[22] |
N. Ichida, T. Sato and M. Linzer,
Imaging the nonlinear ultrasonic parameter of a medium, Ultrasonic Imaging, 5 (1983), 295-299.
doi: 10.1177/016173468300500401. |
[23] |
O. Y. Imanuvilov and M. Yamamoto, Carleman estimate and an inverse source problem for the Kelvin-Voigt model for viscoelasticity, Inverse Problems, 35 (2019), 45pp.
doi: 10.1088/1361-6420/ab323e. |
[24] |
V. Isakov, Inverse Problems for Partial Differential Equations, Applied Mathematical Sciences, 127, Springer, New York, 2006.
doi: 10.1007/0-387-32183-7. |
[25] |
V. K. Ivanov,
On linear problems which are not well-posed, Dokl. Akad. Nauk SSSR, 145 (1962), 270-272.
|
[26] |
B. Kaltenbacher,
An iteratively regularized Gauss-Newton-Halley method for solving nonlinear ill-posed problems, Numer. Math., 131 (2015), 33-57.
doi: 10.1007/s00211-014-0682-5. |
[27] |
B. Kaltenbacher,
Mathematics of nonlinear acoustics, Evol. Equ. Control Theory, 4 (2015), 447-491.
doi: 10.3934/eect.2015.4.447. |
[28] |
B. Kaltenbacher, Periodic solutions and multiharmonic expansions for the Westervelt equation, to appear, Evol. Equ. Control Theory.
doi: 10.3934/eect.2020063. |
[29] |
B. Kaltenbacher and I. Lasiecka,
Global existence and exponential decay rates for the Westervelt equation, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 503-523.
doi: 10.3934/dcdss.2009.2.503. |
[30] |
B. Kaltenbacher and A. Klassen, On convergence and convergence rates for Ivanov and Morozov regularisation and application to some parameter identification problems in elliptic PDEs, Inverse Problems, 34 (2018), 24pp.
doi: 10.1088/1361-6420/aab739. |
[31] |
B. Kaltenbacher, A. Neubauer and O.Scherzer, Iterative Regularization Methods for Nonlinear Ill-Posed Problems, Radon Series on Computational and Applied Mathematics, 6, Walter de Gruyter GmbH & Co. KG, Berlin, 2008.
doi: 10.1515/9783110208276. |
[32] |
V. Kuznetsov, Equations of nonlinear acoustics, Soviet Physics - Acoustics, 16 (1971), 467-470. Google Scholar |
[33] |
M. B. Lesser and R. Seebass,
The structure of a weak shock wave undergoing reflexion from a wall, J. Fluid Mech., 31 (1968), 501-528.
doi: 10.1017/S0022112068000303. |
[34] |
M. J. Lighthill, Viscosity effects in sound waves of finite amplitude, in Surveys in Mechanics, Cambridge, at the University Press, 1956, 250–351. |
[35] |
D. Lorenz and N. Worliczek, Necessary conditions for variational regularisation schemes, Inverse Problems, 29 (2013), 19pp.
doi: 10.1088/0266-5611/29/7/075016. |
[36] |
S. Meyer and M. Wilke,
Optimal regularity and long-time behavior of solutions for the Westervelt equation, Appl. Math. Optim., 64 (2011), 257-271.
doi: 10.1007/s00245-011-9138-9. |
[37] |
V. A. Morozov,
On the solution of functional equations by the method of regularisation, Soviet Math. Dokl., 7 (1966), 414-417.
|
[38] |
M. Muhr, V. Nikolić, B. Wohlmuth and L. Wunderlich,
Isogeometric shape optimization for nonlinear ultrasound focusing, Evol. Equ. Control Theory, 8 (2019), 163-202.
doi: 10.3934/eect.2019010. |
[39] |
A. Neubauer,
On Nesterov acceleration for Landweber iteration of linear ill-posed problems, J. Inverse Ill-Posed Probl., 25 (2017), 381-390.
doi: 10.1515/jiip-2016-0060. |
[40] |
A. Neubauer,
Tikhonov-regularisation of ill-posed linear operator equations on closed convex sets, J. Approx. Theory, 53 (1988), 304-320.
doi: 10.1016/0021-9045(88)90025-1. |
[41] |
A. Neubauer and O. Scherzer,
A convergent rate result for a steepest descent method and a minimal error method for the solution of nonlinear ill-posed problems, Z. Anal. Anwendungen, 14 (1995), 369-377.
doi: 10.4171/ZAA/679. |
[42] |
H. Ockendon and J. R. Ockendon, Waves and Compressible Flow, Texts in Applied Mathematics, 47, Springer-Verlag, New York, 2004.
doi: 10.1007/b97537. |
[43] |
A. Pierce,
Unique identification of eigenvalues and coefficients in a parabolic problem, SIAM J. Control Optim., 17 (1979), 494-499.
doi: 10.1137/0317035. |
[44] |
A. Rieder,
On convergence rates of inexact Newton regularizations, Numer. Math., 88 (2001), 347-365.
doi: 10.1007/PL00005448. |
[45] |
W. Rundell and P. E. Sacks,
Reconstruction techniques for classical inverse Sturm-Liouville problems, Math. Comp., 58 (1992), 161-183.
doi: 10.1090/S0025-5718-1992-1106979-0. |
[46] |
O. Scherzer,
A modified Landweber iteration for solving parameter estimation problems, Appl. Math. Optim., 38 (1998), 45-68.
doi: 10.1007/s002459900081. |
[47] |
T. I. Seidman and C. R. Vogel,
Well-posedness and convergence of some regularisation methods for non-linear ill posed problems, Inverse Problems, 5 (1989), 227-238.
doi: 10.1088/0266-5611/5/2/008. |
[48] |
F. Varray, O. Basset, P. Tortoli and C. Cachard,
Extensions of nonlinear B/A parameter imaging methods for echo mode, IEEE Trans. Ultrasonics, Ferroelectrics, and Frequency Control, 58 (2011), 1232-1244.
doi: 10.1109/TUFFC.2011.1933. |
[49] |
P. J. Westervelt,
Parametric acoustic array, J. Acoustical Soc. Amer., 35 (1963), 535-537.
doi: 10.1121/1.1918525. |
[50] |
M. Yamamoto and B. Kaltenbacher, An inverse source problem related to acoustic nonlinearity parameter imaging, to appear, Time-Dependent Problems in Imaging and Parameter Identification, Springer, 2021. Google Scholar |
[51] |
E. A. Zabolotskaya and R. V. Khokhlov, Quasi-plane waves in the non-linear acoustics of confined beams, Soviet Physics - Acoustics, 15 (1969), 35-40. Google Scholar |
[52] |
D. Zhang, X. Chen and X.-F. Gong,
Acoustic nonlinearity parameter tomography for biological tissues via parametric array from a circular piston source - Theoretical analysis and computer simulations, J. Acoustical Soc. Amer., 109 (2001), 1219-1225.
doi: 10.1121/1.1344160. |
[53] |
D. Zhang, X. Gong and S. Ye,
Acoustic nonlinearity parameter tomography for biological specimens via measurements of the second harmonics, J. Acoustical Soc. Amer., 99 (1996), 2397-2402.
doi: 10.1121/1.415427. |








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